The effect of turbulence on mixing in prototype reaction-diffusion systems

The effect of turbulence on mixing in prototype reaction-diffusion systems is analyzed here in the special situation where the turbulence is modeled ideally with two separated scales consisting of a large-scale mean flow plus a small-scale spatiotemporal periodic flow. In the limit of fast reaction and slow diffusion, it is rigorously proved that the turbulence does not contribute to the location of the mixing zone in the limit and that this mixing zone location is determined solely by advection of the large-scale velocity field. This surprising result contrasts strongly with earlier work of the authors that always yields a large-scale propagation speed enhanced by small-scale turbulence for propagating fronts. The mathematical reasons for these differences are pointed out. This main theorem rigorously justifies the limit equilibrium approximations utilized in non-premixed turbulent diffusion flames and condensation-evaporation modeling in cloud physics in the fast reaction limit. The subtle nature of this result is emphasized by explicit examples presented in the fast reaction and zero-diffusion limit with a nontrivial effecl of turbulence on mixing in the limit. The situation with slow reaction and slow diffusion is also studied in the present work. Here the strong stirring by turbulence before significant reaction occurs necessarily leads to a homogenized limit with the strong mixing effects of turbulence expressed by a rigorous turbulent diffusivity modifying the reaction-diffusion equations. Physical examples from non-premixed turbulent combustion and cloud microphysics modeling are utilized throughout the paper to motivate and interpret the mathematical results.